The Role of Bridge-State Intermediates in Singlet Fission for Donor–Bridge–Acceptor Systems: A Semianalytical Approach to Bridge-Tuning of the Donor–Acceptor Fission Coupling

We describe a semianalytical/computational framework to explore structure–function relationships for singlet fission in Donor (D)–Bridge (B)–Acceptor (A) molecular architectures. The aim of introducing a bridging linker between the D and A molecules is to tune, by modifying the bridge structure, the electronic pathways that lead to fission and to D–A-separated correlated triplets. We identify different bridge-mediation regimes for the effective singlet-fission coupling in the coherent tunneling limit and show how to derive the dominant fission pathways in each regime. We describe the dependence of these regimes on D–B–A many-electron state energetics and on D–B (A–B) one-electron and two-electron matrix elements. This semianalytical approach can be used to guide computational and experimental searches for D–B–A systems with tuned singlet fission rates. We use this approach to interpret the bridge-resonance effect of singlet fission that has been observed in recent experiments.


Many-Electron Spin-Adapted Basis Set
In the first step of the SF process both the initial and final states are singlets, so we consider only singlet states as intermediates for this step (within the CISD formalism, these include single and double excitations). The many-electron states we use describe the D, B or A localization of the excited electron (e) and the hole (h), and are eigenstates of the total spin. They can be represented by linear combinations of singly-and doubly excited N − electron determinants.
The spin eigenfunctions are constructed via the branching diagram method (using the Yamanouchi-Kotani functions). [1][2][3] For The spin-spatial state with a prespecified occupation of spatial orbitals, is constructed by first multiplying a

( )
Xk by a product of the prespecified spatial orbitals , Since we consider only singlet states, we will not use a total-spin label in our notation.
For our active space we can create ( ) α First column: mathematical notation for the spin-adapted many-electron basis states. Second column: spin-spatial multi-electronic states as linear combinations of doubly excited Slater determinants (CTP: correlated Triplet-Pair; CSP: correlated Singlet-Pair).

Diagonal Matrix Elements
In the table below, we give exact expressions for the diagonal elements In our computations we use these exact expressions to compute the Hamiltonian for the reference systems and for the coupling plots. In particular, we compute the 1e and 2e contributions in each equation of Table S3 using the GAMESS-US 11-13 program in the fragment-orbital representation. The main text presents approximate expressions for the lowest-lying states ( Table 1 in the main text). The validity of each approximate expression is verified from the ab initio computations of the different contributions in the exact expressions shown in Table S3.
In the equations of Table S3, ˆk en V − denote Coulombic attractive interactions between the electrons and the th k fragment nuclei (part of the core term in the Hamiltonian).

Off-Diagonal Matrix Elements
In the table below, we present off-diagonal elements, among some of the abovementioned spin-adapted states. As with the diagonal elements, the expressions are evaluated using ab initio computations on the reference systems. In the equations below ----α First column: coupling notation, 1e V denotes coupling dominated by the 1e matrix element (Fock matrix element), while 2e V is a coupling dominated entirely by 2e matrix elements. Second column: notation of the states involved in the corresponding coupling. Third column: exact expressions for the off-diagonal matrix elements as a function of Fock matrix elements and 2e integrals ignoring overlap off-diagonal matrix elements. S12

Effective Coupling Analysis
As mentioned in the main text, we explore the situation where the initial photoexcitation creates a D-localized singlet exciton that can be approximated by (eq. 1 and first row in Table 1 to in (and also to fi , since it has lower energy than in ). We denote this regime of SF as the coherent tunneling regime. Singlet fission will take place when the initial and final states come to resonance at an energy res E . Using standard projection methods, [14][15][16] we approximate the effective coupling for the SF process by    As can be seen from Fig. S1 the largest change in effective coupling comes from the diagonal Hamiltonian elements, as they can cause a change in effective coupling up to four orders of magnitude (OM). In the case of off-diagonal elements (inter-state couplings) the maximum change is limited to one OM. We find that the magnitudes of the 1e V and 2e V do not vary significantly among the reference structures of Fig. 1 (maximum percentage changes and average percentage for 1e V of the order of 40% and 16% , respectively, and for 2e V 33% and 15% , respectively). The maximum coupling magnitude is of the order of 0.1eV .

Generality of the Analytical Model
In the main text and in SI sections 1 and 2, the analytical formulas for energies and What these orbitals are is system-specific and should be deduced from experiment coupled with ab initio computations on the system under study.
The corresponding energies and coupling expressions between diabatic states are also The same holds for the analytical expressions of the off-diagonal matrix elements between the many-electron states (Table S4), e.g., Since the effective coupling is computed exactly by diagonalization of the full Hamiltonian at the initial-to-final state resonance (tunneling) energy (see section 3), the method can treat both asymmetric D-B-A systems and strongly-interacting fragments.